Outline of Today’s Discussion1. Displaying the Order in a Group of Numbers:

2. The Mean, Variance, Standard Deviation, & Z-Scores

3. SPSS: Data Entry, Definition, & Frequency Analysis

Part 1

Displaying the OrderIn a Group of Numbers:

Two Branches of Statistical Methods

• Descriptive statistics– Summarize and describe a group of numbers

such as the results of a research study

• Inferential statistics– Allow researchers to draw conclusions and

inferences that are based on the numbers from a research study, but go beyond these numbers

Basic concepts

• Variables– Characteristics that can take on any number of different

values

• Values (think of this as ‘levels’)– Possible numbers or categories that of a variable can

have

• Scores– A particular person’s value on a variable

Kinds of Variables

• Numeric variables– Scores are numbers that tell you how much there is of the thing

being measured

– Equal-interval variables• Values are numbers that stand for equal amounts of what is

being measured• e.g., a height of 6 feet, an IQ of 138

– Rank-order variables• Values are ranks• e.g., 2nd in the Olympic ski jump, 6th in your graduating class

Kinds of Variables

• Nominal variables– Values are categories– e.g., Hair color of brown, American citizenship

Frequency Tables

• Frequency table– Shows how many times each value was used

for a particular variable– Percentage of scores of each value

• Grouped frequency table– Range of scores in each of several equally sized

intervals

Steps for Making a Frequency Table

1. Make a list of each possible value, from highest to lowest2. Go one by one through the scores, making a mark for each score next

to its value on the list3. Make a table showing how many times each value on your list was

used4. Figure the percentage of scores for each value

A Frequency Table

Travel Time Frequency Percent10 14 9.3

9 15 9.9

8 26 17.2

7 31 20.5

6 13 8.6

5 18 11.9

4 16 10.6

3 12 7.9

2 3 2.0

1 1 0.7

0 2 1.3

A Grouped Frequency Table

Travel Time (min) Frequency Percent

0-5 14 9

8-9 41 27

6-7 44 29

4-5 34 23

2-3 15 10

0-1 3 2

Frequency Graphs• Histogram

– Depicts information from a frequency table or a grouped frequency table as a bar graph

Frequency Graphs• Frequency Polygon

– Depicts the same information by plotting a point to show the frequency of each value or range of values

– Points are joined with a line

Frequency Graphs• Histogram and frequency polygon for the same data

Shapes of Frequency Distributions

• Unimodal– Having one peak

• Bimodal– Having two peaks

• Multimodal– Having two or more peaks

• Rectangular– Having no peaks

Symmetrical vs. Skewed Frequency Distributions

• Symmetrical distribution– Approximately equal numbers of observations above and below

the middle

• Skewed distribution– One side is more spread out that the other, like a tail– Direction of the skew

• Right or left (i.e., positive or negative)• Side with the fewer scores • Side that looks like a tail

Skewed Frequency Distributions

• Skewed right (b)– Fewer scores right of the peak– Positively skewed– Can be caused by a floor effect

• Skewed left (c)– Fewer scores left of the peak– Negatively skewed– Can be caused by a ceiling effect

Potential Pop Quiz Item• Using your imagination, identify a phenomenon that would be well

described by a positively skewed frequency distribution. Draw the positively skewed distribution, and be sure to label your axes. Then briefly explain why the distribution would be positively skewed.

• Repeat the exercise above, but now for a negatively skewed frequency distribution.

Ceiling and Floor Effects

• Ceiling effects– Occur when scores can go no

higher than an upper limit and “pile up” at the top

– e.g., scores on an easy exam, as shown on the right

– Causes negative skew

• Floor effects– Occur when scores can go no

lower than a lower limit and pile up at the bottom

– e.g., household income– Causes positive skew

Kurtosis• Degree to which tails of the distribution are “heavy” or “light”

– heavy tails = many scores at the extreme ends (b)– Light tails = few scores at the extreme ends (c)

leptokurtic

platykurtic

Kurtosis• A perfect, normal distribution zero kurtosis irrespective of its mean or

standard deviation. It is said to be mesokurtic.

• If a random variable’s kurtosis is greater than 0, it is said to be leptokurtic. (peaked, and fat tails i.e., more frequent extreme values than a Gaussian.)

• If its kurtosis is less than 0, it is said to be platykurtic. (flat-ish, and thin tails, i.e., less frequent extreme values than a Gaussian)

• Can you think of a mnemonic for those terms?

The Normal Curve• Often seen in social and behavioral science research and in nature

generally• Also Called “Gaussian” Distribution or Curve• Particular characteristics

– Bell-shaped– Unimodal– Symmetrical– Average tails

The Normal Curve

• Here’s the equation for the Bell Curve:

• Do the equations below look familiar?Y = mX + BY = aX2 + bX + c

• How many parameters in each equation?

                      

Part 2

The Mean, Variance,Standard Deviation,

&Z-Scores

(I LOVE Z-Scores!!!!)

Measures of Central Tendency

• Central tendency = representative or typical value in a distribution

• Mean– Same thing as an average– Computed by

• Summing all the scores (sigma, )• Dividing by the number of scores (N)

M

XN

Measures of Central Tendency

• Mean– Often the best measure of central tendency – Most frequently reported in research articles

• Think of the mean as the “balancing point” of the distribution

Measures of Central Tendency

• Examples of means as balancing points of various distributions– Does not have to be a score exactly at the mean– Note that a score’s distance from the balancing point matters in

addition to the number of scores above or below it

Measures of Central Tendency

• Mode– Most common single number in a distribution– If distribution is symmetrical and unimodal, the

mode = the mean– Typical way of describing central tendency of a

nominal variable

Measures of Central Tendency

• Examples of means and modes

Measures of Central Tendency

• Median– Middle value in a group of scores– Point at which

• half the scores are above• half the scores are below

– Unaffected by extremity of individual scores• Unlike the mean• Preferable as a measure of central tendency when a

distribution has some extreme scores

Measures of Central Tendency

• Steps to computing the median1. Line up scores from highest to lowest2. Figure out how many scores to the middle

• Add 1 to number of scores• Divide by 2

3. Count up to middle score• If there is 1 middle score, that’s the median• If there are 2 middle scores, median is their average

Measures of Variation

• Variation = how spread out scores are• Distributions to the right all have same mean but different

amounts of spread• Variance

– Measure of variation– Average of each score’s squared deviations (differences) from the

mean

Measures of Variation

• Steps to computing the variance– 1. Subtract the mean from each score– 2. Square each deviation score– 3. Add up the squared deviation scores– 4. Divide sum by the number of scores

• Thus, variance is the average (steps 3 & 4) of the squared (step 2) deviations from the mean (step 1)

Measures of Variation

• Standard deviation– Another measure of variation, roughly the average

amount that scores differ from the mean– Used more widely than variance– Abbreviated as “SD”

• To compute standard deviation– Compute variance– Simply take the square root

• SD is square root of variance• Variance is SD squared

Z Scores

• Allow you to describe a particular score in terms of where it fits into the overall group of scores in a distribution– If positive, indicates the number of SDs a score is

above the mean of its own distribution– If negative, indicates the number of SDs a score is

below the mean of its own distribution• Z score of 1.0 is one SD above the mean• Z score of -2.5 is two-and-a-half SDs below the mean• Z score of 0 is at the mean

Z Scores

• When values in a distribution are converted to Z scores, the distribution will have – Mean of 0– Standard deviation of 1

• Useful– Allows variables to be compared to one another even

when they are measured on different scales, have very different distributions, etc.

– Provides a generalized standard of comparison

Z Scores

• To compute a Z score, subtract the mean from a raw score and divide by the SD

• To convert a Z score back to a raw score, multiply the Z score by the SD and then add the mean

Part 3

SPSS: Data Entry, Definition &

Frequency Analysis